Tables of (3+

Harold T. Stokes and Branton J. Campbell,
Department of Physics and Astronomy, Brigham
Young University, Provo, Utah, 84602, USA, stokesh@byu.edu

Sander van Smaalen, University of Bayreuth, Bayreuth, Germany,
smash@uni-bayreuth.de

**ITC-A**:
*International Tables for Crystallography*,
Vol. A, Edited by T. Hahn (Kluwer, Dordrecht, 2002).

**ITC-C**:
T. Janssen, A. Janner, A. Looijenga-Vos, and P. M. de Wolff,
"Incommensurate and commensurate modulated structures,"
*International Tables for Crystallography*,
Vol. C, pp. 907-955. Edited by E. Prince (Kluwer, Dordrecht, 2004).

**JJdW**:
A. Janner, T. Janssen and P. M. de Wolff, "Bravais Classes for
Incommensurate Crystal Phases," *Acta Cryst.* **A39**, 658-666
(1983).

I. P. Orlov and G. Chapuis, "List of (3+1) Dimensional Superspace Groups," superspace.epfl.ch/groups/ (2005).

S. van Smaalen, *Incommensurate Crystallography*, IUCr Monographs on
Crystallography #21 (Oxford University Press, Oxford, 2007).

A. Yamamoto, "Superspace groups for 1D, 2D and 3D modulated structures,"
quasi.nims.go.jp/yamamoto/.
A. Yamamoto, *Acta Crystallogr.*
A52, 509-560, (1996)

*d*=1, 24 Bravais classes, 775 Superspace groups

*d*=2, 83 Bravais classes, 3338 Superspace groups

*d*=3, 215 Bravais classes, 12584 Superspace groups

Any lattice in three dimensions will belong to one of the 14 crystallographic
Bravais classes. Similarly, Bravais classes are used to identify lattice
types in (3+*d*)D superspace.
The identifier of a Bravais class consists of two integers separated
by a period. The first integer denotes the dimension *d* of the
modulations involved (i.e. *d*=1, *d*=2 or
*d*=3). The second number simply enumerates the Bravais
classes for a given dimension. For *d*=1, we list the
Bravais classes in the same order as those listed in ITC-C and JJdW.
For *d*=2, we list the Bravais classes in the same order as those
listed in JJdW. For *d*=3, however, we have corrected several
errors in JJdW (four redundant classes and two missing classes) and
renumbered the *d*=3 classes accordingly. Because a
renumbering was necessary, we also found it convenient to make other
changes to the order of the *d*=3 classes for aesthetic
reasons.

The identifier of a superspace group consists of four integers separated by periods. The first integer refers to the basic space group of the superspace group, and corresponds to one of the 230 three-dimensional space groups in ITC-A. The second and third integers indicate the Bravais class of superspace groups to which the superspace group belongs. The fourth integer enumerates the superspace groups associated with a given basic space group.

ITC-A uses a coordinate triplet to indicate a vector position within
the 3D lattice of the crystal. In treating (3+1)D
superspace-groups, ITC-C uses the symbol "t" to indicate the
phase-shift coordinate. This pattern can be extended to (3+2)
and (3+3) dimensions by using "u" and "v" as
additional phase-shift coordinates. On the *unmixed* basis, which
strictly separates the 3D and phase-coordinate axes,
(*x,y,z,t,u,v*) indicates a point (*x,y,z*) in 3D space as well as
phase shifts of *t*, *u*, and *v* along three
independent modulation directions. Within the superspace
construction, however, an external (3+d)D basis vector can have
non-zero projections onto both 3D and phase-space coordinates.
To distinguish them from unmixed-basis vector components, van Smaalen
uses a
(*x _{s1},x_{s2},x_{s3},x_{s4},x_{s5},x_{s6}*)
notation for vector components defined on the

A (3+*d*)D Bravais class symbol contains a 3D space-group symbol
to identify the three-dimensional symmetry of the external part of the
class, followed by
*d* rationally-independent modulation (**q**) vectors. As in
ITC-A, a 3D space group symbol contains a capital letter (e.g. P, A,
B, C, F, I, R) to indicate a centering type and a point-group symbol
to indicate the Laue group. The 3D external portions of these
Bravais class symbols are identical to those used in JJdW and ITC-C,
except that we place the centering type before, rather than after, the
point-group symbol. In contrast to JJdW, our symbols include a
full *d*-dimensional basis of
**q** vectors rather than including only the generating **q**
vectors. The irrational components of these **q** vectors are
denoted by a=alpha, b=beta and g=gamma, which are then subscripted
with integers (e.g. 1,2,3) to denote
**q** vectors that belong to independent sets of **q** that
are not related to each other via any of the rotational operators
in the group.
Following the conventions of JJdW, we frequently use nonstandard
settings (e.g. Ammm rather than Cmmm) for the 3D portion of a class in
order to obtain modulations vectors with a desired form. One
could also imagine giving precedence to standard 3D space group
settings, and letting the q vectors fall where they may, though such
an approach would be a departure from the past.

Example: 3.128 Immm(a1,b1,0)(-a1,b1,0)(0,0,g2) is a (3+3)D Bravais
class. From "Immm", we learn that the external 3D part of the
class is body-centered orthorhombic. The basis of the internal
part of the class contains a two-dimensional modulation defined by
vectors (a1,b1,0) and (-a1,b1,0), as well as a one-dimensional
modulation defined by vector (0,0,g2). Because the parameters a1
and b1 have the same values in both of the first two **q** vectors,
these two **q** vectors are related to each other via a rotational
operator in Immm, while the third
**q** vector is not.

A (3+*d*)D superspace group symbol can be viewed as a
construction that builds upon the (3+*d*)D Bravais class symbol
to which it belongs. To begin with, the 3D space-group symbol of
the external portion of the Bravais class is replaced by the 3D
space-group symbol of the basic space group (i.e. the external part of
the superspace group symmetry). But the basic space group can be
added in two stages. First, the external point group is chosen
from within the Laue family of the point group of the external Bravais
class, at which point, both the internal and external symmetry
operators are still free of non-lattice translations (i.e. both the
basic space group and the superspace group are symmorphic).
Then, a self-consistent set of non-lattice translations can be added
to the generators of the external point group to obtain a
non-symmorphic basic space group. And finally, a self-consistent
set of internal non-lattice translations can be added to the
point-group generators to obtain a fully general superspace
group.

Historically, superspace group symbols have been designed to reflect the intrinsic (i.e. origin-independent) part of the non-lattice translation associated with each point-group generator. Note that the symbols for the 3D basic space groups, which are taken from ITC-A, already accomplish this via subscripted integers for screw axes and labels like "a" and "n" for glide planes. We chose to adopt and extend the approach of ITC-C, which associates non-zero internal intrinsic translations with lowercase alphabet letters (0, s=1/2, t=1/3, q=1/4, h=1/6) in a one-line symbol. Our symbols are similar to those in ITC-C except that we explicitly include all zeros, even when there are no non-zero translations. In order to appropriately represent all unique intrinsic internal translations, we also use -t=2/3, -q=3/4 and -h=5/6 (van Smaalen, 2007).

For *d* > 1 groups, the
primary difference between our symbols and those of Yamamoto (1996)
is that (1) we display each internal intrinsic translation immediately
after the modulation vector to which it corresponds, rather then
grouping all of the internal translations at the end of the symbol and
(2) we use only the simple symbols s,t,q,h for intrinsic translations
rather than symbols like m,g,d,a,b,c,n that are meant to denote
specific operations in *d*-dimensional internal space. Note that
the internal intrinsic translations are determined from the generators
in the supercentered setting.

Example: 44.3.128.51 I2mm(a1,b1,0)000(-a1,b1,0)000(0,0,g2)0s0 is a (3+3)-dimensional superspace group based on 3D basic space group I2mm and Bravais class Immm(a1,b1,0)(-a1,b1,0)(0,0,g2). The generators of the basic space group, as given in the symbol I2mm, are 2=(x,-y,-z), m=(x,-y,z) and m=(x,y,-z). The corresponding superspace-group generators are (x,-y,-z,t,-u,-v), (x,-y,z,t,-u,v+1/2) and (x,y,-z,t,u,-v+1/2). While the basic space group is symmorphic in this case, there are still some non-zero internal intrinsic translations. Because there are no intrinsic translations associated with u or v coordinates, the translations are displayed as "000" for the first two modulation vectors. Note, however, that the second generator has an intrinsic 1/2 on the v coordinate, which causes the translations of the third modulation vector to be displayed as "0s0".

Detailed information can be obtained about a superspace group by (1) choosing the group from a list, (2) searching for a specific group number, or (3) by using the SSG equivalence finder tool. In each case, the information displayed is identical. We consider a specific example, superspace group 44.3.128.51, in detail. Below, we give each line from the detailed information about this group, followed by an explanation.

**Superspace group:**
44.3.128.51 I2mm(a1,b1,0)000(-a1,b1,0)000(0,0,g2)0s0 [Y:3.3821]

The superspace group number and symbol are
followed by a cross reference to group #3821 in Yamamoto's
*d*=3 superspace group tables. When multiple entries turn
out to be equivalent in Yamamoto's tables, we cross reference all of
them. When a superspace group is missing from Yamamoto's tables, we
display [Y:none]. Note that there are a number of entries in
Yamamoto's *d*=3 tables that do not obey multiplication tables,
and cannot therefore be cross referenced to our tables.

**Bravais class:** 3.128 Immm(a1,b1,0)(-a1,b1,0)(0,0,g2) [JJdW:3.130]

The Bravais class number and symbol are followed by a cross reference
to the JJdW tables of Bravais classes. Because our Bravais-class
numberings only differ from those of JJdW for *d*=3 classes, we
only cross reference JJdW for *d*=3 classes. When multiple
entries turn out to be equivalent in the JJdW tables, we cross
reference all of them. When a class is missing from the JJdW tables,
we display [JJdW:none].

Note that when a superspace group is a member of an enantiomorphic
pair, the other member of the pair is displayed. For example, the
information displayed for 76.1.19.1 P4_{1}(0,0,g)0 contains the
following line:

**Enantiomorph:** 78.1.19.1 P4_{3}(0,0,g)0

**Transformation to supercentered setting:** A1=a1, A2=a2, A3=a3, A4=a4-a5,
A5=a4+a5, A6=a6

Let a1,a2,a3,a4,a5,a6 be the basis vectors of the superspace lattice
in the setting of the 3D basic space group, which is the setting that
puts the external part of each operator into a form that matches the
corresponding entry in ITC-A. Let A1,A2,A3,A4,A5,A6 be the basis
vectors of the superspace lattice in the supercentered setting (i.e. a
setting that eliminates all rational components of the **q**
vectors and simplifies the forms of the **q** vectors). We
explicitly define the transformation between the setting of the basic
space group and the supercentered setting.

Note that when the transformation does not change any of the basis vectors of the lattice (for example, A1=a1, A2=a2, A3=a3, A4=a4, A5=a5, A6=a6), the entry looks like

**Transformation to supercentered setting:** none

**
BASIC SPACE GROUP SETTING**

Setting where the 3D part of the operators are the same as those listed for the basic space group in ITC-A. The internal coordinates are in a primitive setting so that no centering translation involves those coordinates.

**Modulation vectors:** q1=(a1,b1,0), q2=(-a1,b1,0), q3=(0,0,g2)

The modulation vectors in the basic space-group setting are taken directly from the superspace group symbol.

**Centering:** (0,0,0,0,0,0), (1/2,1/2,1/2,0,0,0)

Centering translations that arise in the conventional setting of the basic space group are the same as those in ITC-A. Note that they do not include any internal components.

**Non-lattice generators:** (x,-y,-z,-u,-t,-v); (x,-y,z,-u,-t,v+1/2);
(x,y,-z,t,u,-v+1/2)

The superspace group generators correspond precisely to (and are
listed in the same order as) the non-lattice generators displayed
within the basic space group symbol. Here, I2mm denotes three
generators: (2) a 2-fold rotation about the *a* axis, (m) a
reflection through the *ac* plane, and (m) a reflection through
the
*ab* plane. The lattice translations required to completely generate a
superspace group are not listed here because they are trivial to obtain.

**Non-lattice operators:** (x,y,z,t,u,v); (x,-y,-z,-u,-t,-v);
(x,-y,z,-u,-t,v+1/2); (x,y,-z,t,u,-v+1/2)

In addition to the generators, we list the full set of operators associated with the general Wyckoff position.

Note that if the supercentered setting is the same as the basic space group setting, the reflection conditions are listed here using lower-case letters h,k,l,m,n,p.

**SUPERCENTERED SETTING**

Setting where the **q**-vectors are in a simplified uniform form.
This often results in centering translations that have nonzero
internal coordinates. If the supercentered setting is the same as the
basic space group setting, this section is omitted.

**Modulation vectors:** Q1=(A1,0,0), Q2=(0,B1,0), Q3=(0,0,G2),
where A1=a1, B1=b1, G2=g2

The supercentered setting not only eliminates rational **q**-vector
components, it often simplifies the forms of the **q**
vectors. In this example, multi-component modulation vectors
like (a1,b1,0) and (-a1,b1,0) are separated into single-component
vectors. When the modulation vectors in the supercentered setting are
defined relative to the original modulations vectors in the basic
space-group setting, we use uppercase letters for the supercentered
setting and lowercase letters for the basic space-group setting.

**Centering:** (0,0,0,0,0,0), (1/2,1/2,1/2,0,0,0), (0,0,0,1/2,1/2,0),
(1/2,1/2,1/2,1/2,1/2,0)

Supercentering operations generally add additional centering translations that include components in the internal space. We list the entire group of centering translations.

**Non-lattice generators:** (X,-Y,-Z,T,-U,-V); (X,-Y,Z,T,-U,V+1/2);
(X,Y,-Z,T,U,-V+1/2)

**Non-lattice operators:** (X,Y,Z,T,U,V); (X,-Y,-Z,T,-U,-V);
(X,-Y,Z,T,-U,V+1/2); (X,Y,-Z,T,U,-V+1/2)

The generators and operators from the basic space-group setting are transformed into the supercentered setting and presented in exactly the same order. We use uppercase coordinates (X,Y,Z,T,U,V) for the supercentered setting and lowercase coordinates (x,y,z,t,u,v) for the basic space-group setting. Note that the internal intrinsic translations denoted in the superspace-group symbol are calculated in the supercentered setting.

**Reflection conditions:** HKLMNP:M+N=2n; HKLMNP:H+K+L=2n; H0LM0P:P=2n

The list of reflection conditions includes both the centering translations of the Bravais class and any non-lattice translations possessed by non-symmorphic groups. The list of conditions is always minimal, which means that any other reflections conditions that can be defined will not add any extinctions to those already created by the conditions in the list. When a supercentered setting is available, we only express the reflection conditions in the supercentered setting, and use uppercase letters (H,K,L,M,N,P) in the conditions. When no supercentered setting is available, we express the reflection conditions in the basic space-group setting, and use lowercase letters (h,k,l,m,n,p) in the conditions. We use "p" instead of "o" in the reflection conditions because "o" and "O" appear very similar to zero.

Due to their dependence on the intrinsic translational components of
the generators, space-group symbols can be ambiguous when multiple
groups can have the same intrinsic translations. This ambiguity
arises only in centered groups, where one can combine a non-lattice
generator with a centering translation to produce an equivalent
generator that may have different intrinsic translational components.
As a classic example, there are eight ways to combine the
(1/2,1/2,1/2) centering translation with the three two-fold rotation
generators of space group #23 (I222), resulting in eight different
sets of intrinsic translations and eight possible symbols for the
group: I222, I222_{1}, I22_{1}2,
I22_{1}2_{1}, I2_{1}22,
I2_{1}22_{1}, I2_{1}2_{1}2,
I2_{1}2_{1}2_{1}. Surprisingly, the same
procedure applied to space group #24
(I2_{1}2_{1}2_{1}) produces the same eight
symbols. Thus, the traditional choice of I222 as the symbol for space
group #23 and I2_{1}2_{1}2_{1} as the symbol
for space group #24 is not grounded purely in the intrinsic
translations, but requires other considerations. The only other 3D
space groups subject to this problem are I23 and I2_{1}3. For
superspace groups, which have a large number of rather complex
centering types, this problem is greatly compounded. While the
present system of symbols is still based on intrinsic translations, we
have endeavored to define conventions that ensure that distinct groups
are given distinct symbols, and that the symbol assigned to each group
is as simple as possible. We describe these conventions below.

**Generators of basic space group**

The primary restriction that we place on the lattice translations to be combined with any given generator are that the resulting intrinsic translation must strictly match the symbol of the basic space group in ITC-A. In cases where ITC-A gives more than one setting for a space group, we use the following setting conventions: (1) monoclinic: cell choice 1, (2) trigonal: hexagonal axes, (3) origin choice 2.

The symbol for the basic space group contains information about the
generating symmetry operators. For example, Pmc2_{1} has three
generators: (1) m=mirror reflection through (100), (2) c=glide
reflection through (010) with an intrinsic translation along
[001], and (3) 2_{1}=screw rotation along [001] with an intrinsic
translation also along [001]. The part of the space group symbol
corresponding to a given generator does not uniquely identify a
symmetry operator of the space group, but rather identifies a class of
operators (e.g. all of the c-glide planes perpendicular to the
*b* axis). Sometimes, the symbol of the point-group generator
doesn't uniquely specify the orientation of the operator (e.g. the
first "2" in P422 could represent a rotation around either the
*a* or *b* axis). Because the intrinsic translation of the
generator is often influenced by its orientation, we list the point
groups with generator ambiguities below, together with the generators
that we have chosen for them.

422 | (-y,x,z) (x,-y,-z) (-y,-x,-z) |

4mm | (-y,x,z) (-x,y,z) (y,x,z) |

-42m | (y,-x,-z) (x,-y,-z) (y,x,z) |

-4m2 | (y,-x,-z) (-x,y,z) (-y,-x,-z) |

4/mmm | (-y,x,z) (x,y,-z) (-x,y,z) (y,x,z) |

312 | (-y,x-y,z) (x,y,z) (-y,-x,-z) |

321 | (-y,x-y,z) (-x,-x+y,-z) (x,y,z) |

3m1 | (-y,x-y,z) (x,x-y,z) (x,y,z) |

31m | (-y,x-y,z) (x,y,z) (y,x,z) |

622 | (x-y,x,z) (-x,-x+y,-z) (-y,-x,-z) |

6mm | (x-y,x,z) (x,x-y,z) (y,x,z) |

-6m2 | (-x+y,-x,-z) (x,x-y,z) (-y,-x,-z) |

-62m | (-x+y,-x,-z) (-x,-x+y,-z) (y,x,z) |

6/mmm | (x-y,x,z) (x,y,-z) (x,x-y,z) (y,x,z) |

23 | (-x,-y,z) (z,x,y) |

m-3 | (x,y,-z) (-z,-x,-y) |

432 | (-y,x,z) (z,x,y) (-y,-x,-z) |

-43m | (y,-x,-z) (z,x,y) (y,x,z) |

m-3m | (x,y,-z) (-z,-x,-y) (y,x,z) |

The translational components in each generating operator are generally chosen so that they are positive and less than 1. However, we often find it necessary to use equivalent translation components (related by a lattice translation) outside this range in order to ensure that the generators operate in a way that exactly matches the space group symbol. The ITC-A does not generally concern itself with this level of detail. For example, space group 100 P4bm has both mirror and glide planes parallel to the [110] directions. In the symbol, the third generator is a mirror reflection, though the symmetry operator listed in ITC-A is (y+1/2,x+1/2,z), which has an intrinsic translation of (1/2,1/2,0) and is therefore actually a glide reflection. Because the operator (y+1/2,x-1/2,z) has an intrinsic transition of (0,0,0), and is therefore a true mirror reflection, we use it instead. A similar situation arises in space groups with centered lattices. For example, the third generator in 67 Cmma is a glide reflection (x+1/2,y,-z). In ITC-A, this operator is listed as (x,y+1/2,-z) which would be the third generator for Cmmb (an alternate symbol for Cmma). In order to strictly represent the symbol Cmma, we use (x+1/2,y,-z) as the third generator rather than (x,y+1/2,-z).

**Transformation to supercentered setting**

We specify the transformation of the superspace group from the basic space group setting to the supercentered setting for each Bravais class. We use this same transformation for every superspace group that belongs to the same Bravais class.

**Choice of internal intrinsic translations**

We use Eq. (9.8.3.5) in ITC-C to calculate the origin-invariant part of
the translation associated with each operator in the superspace group.
We call this the intrinsic translation. If we denote an operator by
{*R*|*v*}, then, using a simplified notation, the intrinsic
translation *vi* for this operator is given by

*vi* = (1/*n*) * sum(*m*=1,*n*)
(*R*^{m})**v*,

where *n* is the order of *R*, i.e.,
*R*^{n}=1.

The intrinsic translations are not unique since adding a lattice
translation to *v* can change *vi*. The symbol for the
superspace group is also not unique, since the nonzero internal
components of the intrinsic translations of the generators, denoted by
the symbols (0,s,t,q,h), become part of the symbol itself.

For each group, we generate every possible symbol for the superspace
group and then choose the one with the "nicest" appearance. We
generate candidate symbols by adding a variety of different lattice
translations (both centering translations and conventional lattice
translations) to the translational part *v* of each generator. In
order to strictly respect the basic space group symbol, we only
explore lattice translations which preserve the external parts (mod 1)
of the intrinsic translations *vi* in the basic space group
setting. While the set of lattice translations is infinite, the set
that needs to be tested is limited by the fact that adding the cyclic
order *n* of *R* to any component of *v* does not
affect *vi*. We choose the "nicest"-looking symbol by applying
the following criteria to the internal translational components, in
order of priority:

(1) minimum number of negative components

(2) maximum number of zero components

(3) minimum value of the maximum denominator among the components

(4) smallest denominators occur first

(5) smallest numerators occur first

Sometimes, a number of superspace groups will all share the same candidate symbols, and therefore, the same “nicest” symbol. We say that these superspace groups are “degenerate” with respect to their symbols. We find that we can almost always lift this degeneracy by assigning the nicest symbol to the first group in the degenerate set, and then simply requiring that the external translational components of corresponding generators be exactly identical (not just equivalent mod 1) for each of the groups in the degenerate set.

**Example 1**: 48.2.51.12 Pnnn(1/2,b,g)q0q(1/2,-b,g)qq0. The
generators in the supercentered setting are
(-X,Y+1/2,Z+1/2,T+1/4,U+1/4), (X+1/4,-Y,Z+1/2,-T,U+1/4), and
(X+1/4,Y+1/2,-Z,T+1/4,-U). There are four centering translations for
this Bravais class: (0,0,0,0,0), (1/2,0,0,0,1/2), (0,0,0,1/2,1/2),
(1/2,0,0,1/2,0). We find the following possible internal intrinsic
translations for each generator:

Original generator | Lattice translation | New generator | IIT | IIT Symbols |

(-X,Y+1/2,Z+1/2,T+1/4,U+1/4) | (0,0,0,0,0) | (-X,Y+1/2,Z+1/2,T+1/4,U+1/4) | 1/4,1/4 | q,q |

(1/2,0,0,0,-1/2) | (-X+1/2,Y+1/2,Z+1/2,T+1/4,U-1/4) | 1/4,-1/4 | q,-q | |

(0,0,0,-1/2,-1/2) | (-X,Y+1/2,Z+1/2,T-1/4,U-1/4) | -1/4,-1/4 | -q,-q | |

(1/2,0,0,-1/2,0) | (-X+1/2,Y+1/2,Z+1/2,T-1/4,U+1/4) | -1/4,1/4 | -q,q | |

(X+1/4,-Y,Z+1/2,-T,U+1/4) | (0,0,0,0,0) | (X+1/4,-Y,Z+1/2,-T,U+1/4) | 0,1/4 | 0,q |

(-1/2,0,0,0,-1/2) | (X-1/4,-Y,Z+1/2,-T,U-1/4) | 0,-1/4 | 0,-q | |

(0,0,0,1/2,-1/2) | (X+1/4,-Y,Z+1/2,-T+1/2,U-1/4) | 0,-1/4 | 0,-q | |

(-1/2,0,0,1/2,0) | (X-1/4,-Y,Z+1/2,-T+1/2,U+1/4) | 0,1/4 | 0,q | |

(X+1/4,Y+1/2,-Z,T+1/4,-U) | (0,0,0,0,0) | (X+1/4,Y+1/2,-Z,T+1/4,-U) | 1/4,0 | q,0 |

(-1/2,0,0,0,1/2) | (X-1/4,Y+1/2,-Z,T+1/4,-U+1/2) | 1/4,0 | q,0 | |

(0,0,0,-1/2,1/2) | (X+1/4,Y+1/2,-Z,T-1/4,-U+1/2) | -1/4,0 | -q,0 | |

(-1/2,0,0,-1/2,0) | (X-1/4,Y+1/2,-Z,T-1/4,-U) | -1/4,0 | -q,0 |

Based on four symbol sets for the first generator and two unique symbol sets for the second and third generators, we have sixteen possible symbols for this superspace group. Of these, the nicest looking one is the one without any minus signs in the symbols.

**Example 2**: 100.2.68.12 P4bm(a,a,0)00s(-a,a,0)000. In this
case, though there are no centering translations (thus no distinction
between the basic space-group setting and the supercentered setting),
we can still obtain different symbols by using integer lattice
translations with the generators.

Original generator | Lattice translation | New generator | Intrinsic translation | IIT Symbols |

(-y,x,z,-u,t) | (0,0,0,0,0) | (-y,x,z,-u,t) | (0,0,0,0,0) | 0,0 |

(-x+1/2,y+1/2,z,u+1/2,t+1/2) | (0,0,0,0,0) | (-x+1/2,y+1/2,z,u+1/2,t+1/2) | (0,1/2,0,1/2,1/2) | s,s |

(0,0,0,0,-1) | (-x+1/2,y+1/2,z,u+1/2,t-1/2) | (0,1/2,0,0,0) | 0,0 | |

(y+1/2,x-1/2,z,t+1/2,-u+1/2) | (0,0,0,0,0) | (y+1/2,x-1/2,z,t+1/2,-u+1/2) | (0,0,0,1/2,0) | s,0 |

In this case, there are two possible symbols. We choose the symbol with the greatest number of zeros.

**Example 3**: 47.2.36.60 Pmmm(1/2,b1,1/2)000(1/2,0,g2)000 is the
first member of a set of groups with the same nicest symbol
("degenerate set"). For brevity's sake, let us just consider the first
generator: (-x,y,z,-x+t,-x+u) in the basic space group setting and
(-X,Y,Z,T,U) in the supercentered setting with each of three centering
translations: (1/2,0,0,1/2,1/2), (0,0,1/2,1/2,0), and
(1/2,0,1/2,0,1/2).

Lattice translation (supercentered setting) | New generator (supercentered setting) |
New generator (basic space group setting) | Intrinsic translation (supercentered setting) | IIT Symbols |

(0,0,0,0,0,0) | (-X,Y,Z,T,U) | (-x,y,z,-x+t,-x+u) | (0,0,0,0,0) | 0,0 |

(1/2,0,0,1/2,1/2) | (-X+1/2,Y,Z,T+1/2,U+1/2) | (-x+1,-y,z,-x+t+1,-x+u+1) | (0,0,0,1/2,1/2) | s,s |

(0,0,1/2,1/2,0) | (-X,Y,Z+1/2,T+1/2,U) | (-x,y,z+1,-x+t+1,-x+u) | (0,0,1/2,1/2,0) | s,0 |

(1/2,0,1/2,0,1/2) | (-X+1/2,Y,Z+1/2,T,U+1/2) | (-x+1,y,z+1,-x+t+1,-x+u+1) | (0,0,1/2,0,1/2) | 0,s |

Note that although the last two centering translations change the external part of the intrinsic translational components in the supercentered setting, they do not affect the external part (mod 1) of the intrinsic translational components in the basic space group setting. So we are allowed to consider them because they still agree with the basic space group symbol. Of course, we have chosen the symbol with the greatest number of zeros. Once we choose the symbol for 47.2.36.60, the symbols for each of the other three groups in the degenerate set are also determined since (0,0,0,0,0,0) is the only lattice translation that won't alter the external translational components of the generators. The four superspace groups in this degenerate set are:

47.2.36.60 Pmmm(1/2,b1,1/2)000(1/2,0,g2)000

47.2.36.61 Pmmm(1/2,b1,1/2)000(1/2,0,g2)0s0

47.2.36.62 Pmmm(1/2,b1,1/2)000(1/2,0,g2)s00

47.2.36.63 Pmmm(1/2,b1,1/2)000(1/2,0,g2)ss0

**Example 4**: 16.3.137.108 P222(0,b,g)000(a,0,g)000(a,b,0)000 and
16.3.137.109 P222(0,b,g)000(a,0,g)000(a,b,0)00s. There is one
centering translation (0,0,0,1/2,1/2,1/2) associated with this Bravais
class. We find the following possible internal intrinsic translations
for each generator:

Original generator 16.3.137.108 | Lattice translation | New generator | Intrinsic translation | IIT Symbols |

(X,-Y,-Z,T,-U,-V) | (0,0,0,0,0,0) | (X,-Y,-Z,T,-U,-V) | (0,0,0,0,0,0) | 0,0,0 |

(0,0,0,1/2,1/2,1/2) | (X,-Y,-Z,T+1/2,-U+1/2,-V+1/2) | (0,0,0,1/2,0,0) | s,0,0 | |

(-X,Y,-Z,-T,U,-V) | (0,0,0,0,0,0) | (-X,Y,-Z,-T,U,-V) | (0,0,0,0,0,0) | 0,0,0 |

(0,0,0,1/2,1/2,1/2) | (-X,Y,-Z,-T+1/2,U+1/2,-V+1/2) | (0,0,0,0,1/2,0) | 0,s,0 | |

(-X,-Y,Z,-T,-U,V) | (0,0,0,0,0,0) | (-X,-Y,Z,-T,-U,V) | (0,0,0,0,0,0) | 0,0,0 |

(0,0,0,1/2,1/2,1/2) | (-X,-Y,Z,-T+1/2,-U+1/2,V+1/2) | (0,0,0,0,0,1/2) | 0,0,s |

Original generator 16.3.137.109 | Lattice translation | New generator | Intrinsic translation | IIT Symbols |

(X,-Y,-Z,T,-U+1/2,-V) | (0,0,0,0,0,0) | (X,-Y,-Z,T,-U+1/2,-V) | (0,0,0,0,0,0) | 0,0,0 |

(0,0,0,1/2,-1/2,1/2) | (X,-Y,-Z,T+1/2,-U,-V+1/2) | (0,0,0,1/2,0,0) | s,0,0 | |

(-X,Y,-Z,-T,U,-V+1/2) | (0,0,0,0,0,0) | (-X,Y,-Z,-T,U,-V+1/2) | (0,0,0,0,0,0) | 0,0,0 |

(0,0,0,1/2,1/2,-1/2) | (-X,Y,-Z,-T+1/2,U+1/2,-V) | (0,0,0,0,1/2,0) | 0,s,0 | |

(-X,-Y,Z,-T+1/2,-U,V) | (0,0,0,0,0,0) | (-X,-Y,Z,-T+1/2,-U,V) | (0,0,0,0,0,0) | 0,0,0 |

(0,0,0,-1/2,1/2,1/2) | (-X,-Y,Z,-T,-U+1/2,V+1/2) | (0,0,0,0,0,1/2) | 0,0,s |

Observe that we obtain the same set of candidate symbols for each
group. Thus, our rules of respecting the basic space-group symbol and
of maintaining the same external translational components across an
entire degenerate set of groups have not entirely prevented symbol
collisions. But the collisions are very rare and are easily resolved
by manually assigning appropriate generators. The symmorphic group
16.3.137.108 clearly has a greater claim on the symbol with all-zero
translations. For 16.3.137.109, we simply choose the next-nicest
symbol which includes one "s". All total, there are seven pairs of
groups (listed below) that still have identical sets of candidate
symbols. In each case, one of the groups is symmorphic and receives
the symbol with no translational components, while the other group
gets the next-nicest symbol available. It is interesting that the
three-dimensional internal portion of each of these (3+3)D superspace
groups is identical to one of I222,
I2_{1}2_{1}2_{1}, I23 or I2_{1}3.

16.3.137.109 P222(0,b,g)000(a,0,g)000(a,b,0)00s

21.3.142.106 C222(0,b,g)000(a,0,g)000(a,b,0)00s

22.3.144.38 F222(0,b,g)000(a,0,g)000(a,b,0)00s

23.3.141.16 I222(0,b,g)000(a,0,g)000(a,b,0)00s

195.3.210.8 P23(0,b,b)00(b,0,b)00(b,b,0)s0

196.3.212.6 F23(0,b,b)00(b,0,b)00(b,b,0)s0

197.3.211.4 I23(0,b,b)00(b,0,b)00(b,b,0)s0

**Symbols in ITC-C**

Symbols for the (3+1)-D superspace groups have been listed in ITC-C. There are only eleven cases where the symbols that follow from our conventions disagree with the ITC-C symbols. In eight of these cases, our method found nicer symbols:

Group | ITC-C Symbol | Symbol Using Our Method |

35.1.14.5 | Cmm2(1,0,g)s0s | Cmm2(1,0,g)s00 |

36.1.14.4 | Cmc2_1(1,0,g)s0s | Cmc2_1(1,0,g)s00 |

37.1.14.4 | Ccc2(1,0,g)s0s | Ccc2(1,0,g)s00 |

42.1.18.5 | Fmm2(1,0,g)s0s | Fmm2(1,0,g)s00 |

99.1.20.6 | P4mm(1/2,1/2,g)0ss | P4mm(1/2,1/2,g)00s |

101.1.20.4 | P4_2cm(1/2,1/2,g)0ss | P4_2cm(1/2,1/2,g)00s |

123.1.20.6 | P4/mmm(1/2,1/2,g)00ss | P4/mmm(1/2,1/2,g)000s |

132.1.20.4 | P4_2/mcm(1/2,1/2,g)00ss | P4_2/mcm(1/2,1/2,g)000s |

In three of these cases, our method could not obtain the ITC-C symbol without using external intrinsic translations that were inconsistent with the basic space group symbol.

Group | ITC-C Symbol | Symbol Using Our Method |

104.1.20.3 | P4nc(1/2,1/2,g)qq0 | P4nc(1/2,1/2,g)qqs |

106.1.20.3 | P4_{2}bc(1/2,1/2,g)qq0 | P4_{2}bc(1/2,1/2,g)qqs |

126.1.20.3 | P4/nnc(1/2,1/2,g)q0q0 | P4/nnc(1/2,1/2,g)q0qs |

We do not advocate changing the symbols established in ITC-C. Therefore, we propose to continue using the ITC-C symbols, and not the new symbols generated by our method.

Comparison with Yamamoto's tables of superspace groups

Comparision with Orlov and Chapuis's tables of superspace groups

Comparison with JJdW's tables of Bravais classes